7Short Range Repulsive Interactions When two non-bonded atoms approach each each other, at some distance overlap of the occupied orbitals results in electrostatic repulsion between the electrons of those atoms.This repulsive energy acts over a very short range, but goes up very sharply when that range is violated.The repulsion goes up as 1/r12. It is important only when atoms are in very close proximity, but then it becomes very important
8Because this repulsive term rises so sharply as distance decreases it is sometimes reasonable to think of atoms as hard spheres, like small pool balls, defined by van der Waals radii and surfaces.When two atoms approach each other their van der Waals surfaces make contact when the distance between them reaches the sum of their van der Waals radii.Here we are assuming that bonds do not form. When bonds form van der Waals radii are violated.The smallest distance between two non-bonded atoms is the sum of the van der Waals radii of the two atoms.
9The van der Waal radius of carbon is evident from the spacing between the layers in graphite. The distance between atoms in different layers of graphite is never less than twice the van der Waals radius of carbon (2 x 1.7 = 3.4 Å).The atoms within a graphite layer are covalently linked and so are in violation of the van der Waals radius.
15Due to non-uniform distributions of positive and negative charges on the various atoms, many molecules have dipole moments.Such is the case with polar compounds like water (H2O), where electron density is shared unequally between atoms.The physical chemist Peter J. W. Debye was the first scientist to study molecular dipoles extensively.
16Dipole moment (): = ql (1)where q : chargel : distance of positive and negative chargesDipole energy:(2)where 0 : permittivity of vacuum (= F/m)r : relative permittivity or dielectric constant of the medium where the charges are located in.
17Permanent dipoles: For molecules there are three types of dipoles: These occur when two atoms in a molecule have substantially different electronegativity: One atom attracts electrons more than another, becoming more negative, while the other atom becomes more positive. A molecule with a permanent dipole moment is called a polar molecule.
18Instantaneous dipoles: These occur due to chance when electrons happen to be more concentrated in one place than another in a molecule, creating a temporary dipole.(http://www.chemprofessor.com/imf.htm)
20Induced dipoles:These can occur when one molecule with a permanent dipole repels another molecule's electrons, inducing a dipole moment in that molecule. A molecule is polarized when it carries an induced dipole. See induced-dipole attraction.
21Dipole Moment (Debyes) (measured in the gas phase) Dipole moment values of some typical gas phase in debye units are:CompoundDipole Moment (Debyes)NaCl9.0(measured in the gas phase)CH3Cl1.87H2O1.85NH31.47CO2CCl4
23A dipole that is close to a positive or negative ion will orient itself so that the end whose partial charge is opposite to the ion charge will point toward the ion.This kind of interaction is very important in aqueous solutions of ionic substances.H2O is a highly polar molecule, so that in a solution of sodium chloride, the Na+ ions will be enveloped by a shell of water molecules with their oxygen-ends pointing toward these ions, while H2O molecules surrounding the Cl– ions will have their hydrogen ends directed inward.
24The interaction energy can be constructed from the Coulomb interactions between the bare charge Q and the dipolar charges q:r1r2(3)
25whereQ : charge of ionr : distance between ion and dipole0 : permittivity of vacuum (= F/m)r : relative permittivity or dielectric constant of the medium where the charges are located inl : length of dipole moment vector : the angle between the dipole moment vector and the vector connecting the ion with the dipoleWhen the dipole is sufficiently far away from the charge (r >> l), we can approximate:
27When deriving eq. (4) we assumed the dipole to be sufficiently far from the charge (r >> l). At about r < 2l the approximation deviate more than 10% from the exact result.Taking into account the finite size of atoms and molecules, eq. (4) is actually an excellent approxi- mation for interactions between ions and small polar molecules at all physically relevant distances.In the case of larger molecules, where the charges comprising the dipole may be several ångströms apart, we need to exercise good judgement whether to use eq. (4) or not.
30As two dipoles approach each other, they will tend to orient themselves so that their oppositely-charged ends are adjacent.Two such arrangements are possible: the dipoles can be side by side but pointing in opposite directions, or they can be end to end.It can be shown that the end-to-end arrangement gives a lower potential energy.Dipole-dipole attraction is weaker than ion-dipole attraction, but it can still have significant effects if the dipole moments are large. The most important example of dipole-dipole attraction is hydrogen bonding.
31for r > 3 l:(5)1 and 2 are the polar orientation angles of 1 and 2, respectively, and is the azimuthal orientation angle of 2 in reference to 1
33Dipole-dipole interaction is comparatively weak (for dipole moment of 1 Debye at 0.35 nm in vacuum, the interaction is already weaker than kT).In certain molecules (small size and large dipole moment O-H, N-H, and F-H), dipole-dipole interaction can lead to short range association in liquid (part of H-bond).Dipole-dipole interaction is strongest when the two dipoles mutually orient themselves in line.
34At large separation or in a medium of high , when interaction falls below kT, dipoles can now rotate more freely.The angle averaged potentials are not zero because of Boltzmann weighting factor, the energy (Keesom interaction or orientation interaction) becomes(6)for
35Ion-Induced Dipole Interactions Polarizability ()Polarizability is the ease of distortion of the electron cloud of a molecular entity by an electric field (such as that due to the proximity of a charged reagent).It is experimentally measured as the ratio of induced dipole moment (ind) to the field E which induces it:(7)
36The most significant induced dipole effects result from nearby ions, particularly cations (positive ions).Nearby ions can distort the electron clouds even in polar molecules, thus temporarily changing their dipole moments.The larger ions (especially negative ones such as SO22– and ClO42–) are highly polarizable, and the dipole moments induced in them by a cation can play a dominant role in compound formation.
37Dipole-Induced Dipole Interactions (induction or Debye interaction) (http://textbook.s-anand.net/ncert/class-11/chemistry/5-states-of-matter)
38A permanent dipole can induce a temporary one in a species that is normally non-polar, and thus produce a net attractive force between the two particles.This attraction is usually rather weak, but in a few cases it can lead to the formation of loosely-bound compounds.This effect explains the otherwise surprising observation that a wide variety of neutral molecules such as hydrocarbons, and even some of the noble gas elements, form stable hydrate compounds with water(8)
39Dispersion or London Forces (http://www.chem.ufl.edu/~itl/4411/lectures/lec_g.html)Noble gas elements and completely non-polar molecules such as H2 and N2 can be condensed to liquids or solids.There must be another source of attraction between particles that does not depend on the existence of permanent dipole moments in either particle.
40A molecule is “nonpolar” the time-averaged dipole moment is zero. On a very short time scale, however, the electron must be increasingly localized.As a consequence, there is no guarantee that the distribution of negative charge around the center of an atom will be perfectly symmetrical at every instant; every atom therefore has a weak, fluctuating dipole moment that is continually disappearing and reappearing in another direction.Although these extremely short-lived fluctuations quickly average out to zero, they can still induce new dipoles in a neighboring atom or molecule, which helps sustain the original dipole and gives rise to a weak attractive force known as the dispersion or London force
41Dispersion is applicable to all atoms or molecules (unlike Keesom or Debye interaction). It is responsible for certain phenomena in macro- scopic scale (adhesion, surface tension, physical adsorption, wetting, properties of gases and liquid, structures of condensed macromolecules,... ).It is a long range force that can be effective at large distance (>10 nm) to interatomic spacings.Dispersion is non-additive. The dispersion of two molecules is affected by the presence of the third molecules
42Fritz London (1937) proposed a theory based on quantum mechanics to explain dispersion (9)where I is the first ionization potential I = h
43van der Waals Interaction Many kinds of molecules possess permanent dipole moments, so liquids and solids composed of these species will be held together by a combination of dipole-dipole, dipole-induced dipole, and dispersion forces.These weaker forces (that is, those other than coulombic attractions) are known collectively as van der Waals forces.These are short-ranged and weak interactions existing between all types of atoms and molecules.
44All atoms and molecules, even non-polar and uncharged ones, exert attractive forces on each other. This is a result of the atomic polarizability 0 of atoms.The constant motion of electrons in atoms results in the fact that at any given instant in time, any atom actually has a finite electric dipole moment.Despite the quantum-mechanical uncertainty of position, even particles as light as electrons have to occupy some region of space at a given time.In the absence of an external influence, the average dipole moment of an atom is zero.
45In general, the van der Waals forces arise from three different contributions: orientation or Keesom interaction;induction or Debye interaction; anddispersion or London interactionThus, in general the total van der Waals energy is given by(10)
46The dispersion term is the most important of the three contributions, as it is always present, regardless whether permanent dipoles take part in the interaction or not.Moreover, usually the dispersion term is also the strongest, contributing around % to the total van der Waals interaction energy.A notable exception to this is water, where in fact the Keesom interaction dominates (about 70% of the total interaction energy)
48Intermolecular forces are responsible most properties of all the phases: Gas: Vapor pressure, critical point, and boiling pointLiquid: viscosity, diffusion, and surface tension.Solid: melting and sublimation.
49The calculation of the potential energy involves assumptions concerning the nature of attraction and repulsion between molecules.Intermolecular interaction is the result of both short- and long-range effects.Electrostatic, induction, and dispersion effects are examples of long range interactions.
50In these cases, the energy of interaction is proportional to some inverse power of intermolecular separation. Electrostatic interactions result from the static charge distribution between molecules.The effect can be either attractive or repulsive and it is exclusively pairwise additive.Induction effects are always attractive, resulting from the distortions caused by the molecular fields of neighbouring molecules
51The specification of intermolecular potential, representing interaction between molecules, is a critical step in Molecular Dynamics simulations.Generally, a two-body potential of the form U(rij) is used, where rij is the distance between the centers of molecules i and j.This form neglects multibody interactions.Once the potential is prescribed, the intermolecular force is obtained from Fij = -dU(rij)/drij.In the following, we will discuss intermolecular potential models.
52Hard-Sphere Potential In this model, the molecules move freely and do not interact with one another except when they collide. The intermolecular potential is given by(11)Here (σ) is the diameter of the molecule.Thus the molecules exert force on one another only when they collide
53Intermolecular potential for the hard-sphere model
54Square-Well Potential The square-well potential is the simplest inter-molecular potential that is capable of representing the properties of liquids(12)where is some multiple of the hard-sphere diameter and is a measure of the attractive interaction.The square-well potential represents a mathematically idealised model of molecular interactions.
56Yukawa PotentialThe square-well potential can be made more realistic by changing the variation of attractive interactions. There have been many such variations of which the Yukawa potential is an important example.(13)where is an attractive term (depth of potential well), is the hard-sphere diameter and is an adjustable parameter.The inverse power dependence of this potential means that it can be applied to ionic systems.
57Hard-core Yukawa potential with various interaction ranges (Naresh and Singh, 2000)
58Lennard-Jones Potential: The Lennard-Jones potential (also referred to as the L-J potential, 6-12 potential, or 12-6 potential) is a mathematically simple model that approximates the interaction between a pair of neutral atoms or molecules.(14)where is the depth of potential well, is the hard-sphere diameter (the finite distance at which the inter-particle potential is zero), and r is the distance between the particles.
61The parameter σ is the zero energy separation distance, and defines a molecular length scale related to the particle diameter, while ε is the minimum energy and controls the strength of the interaction.As an example, σ= 0.41 nm, and ε/kB = 221K for xenon, where kB is the Boltzmann's constant.For a pure substance, σ equals the particle diameter.The parameter σ is also related to the critical volume of the fluid, while ε to its critical temperature.This dependence is particularly strong for small molecules.
62For simulations involving two fluids, σ and ε are expressed using the Lorentz-Berthelot mixing rules (Allen and Tildesley, 1984), as:(15)(16)
63Intermolecular ForceThe force between the two L-J molecules is given by(17)By convention, repulsive (short-range) forces are positive while attractive (long-range) forces are negative. i.e.,(18)